A Nonlinear Theory of Generalized Functions

丛书	Lecture Notes in Mathematics
出版社	Springer Berlin / Heidelberg
ISSN	0075-8434 (Print) 1617-9692 (Online)
卷	Volume 1421/1990
DOI	10.1007/BFb0089552
版权	1990
ISBN	978-3-540-52408-3
学科分类	数学和统计学
SpringerLink Date	2006年11月14日

INTRODUCTION
CHAPTER I - GENERALIZED FUNCTIONS ON AN OPEN SUBSET OF E~ ........
§ I I The original definition ................................. I
§ I 2 An elementary definition ................................ 6
§ I 3 Local properties, restrictions and compositions ........ 21
§ 1 4 Nonlinear properties of generalized functions .......... 27
§ I 5 Pointvalues and integration theory ..................... 30
§ I 6 Association processes .................................. 34
§ I 7 Topologies on ~ and ~(~) ............................. 40
§ I 8 The subspace ~s (~) ................................... 45
§ I 9 Heaviside generalized functions ........................ 49
§ 1.10 Generalized solutions of algebraic differential equations
and classical solutions ................................ 55
APPENDIX-A survey on products of distributions .................. 65
CHAPTER 2 - GENERALIZED FUNCTIONS ON AN ARBITRARY SUBSET OF En .69
§2.1
§2.2
§2.3
§2.4
§2.5
§ 2.6
Generalized functions on the closure of an open set .... 69
Essential facts concerning C ~ functions in the sense
of Whitney ............................................. 70
Generalized functions on an arbitrary subset of E n .... 72
Whitney's extension theorem for generalized functions..74
Borel's theorem for generalized functions .............. 75
Extension of a generalized function defined on a halfspace
.................................................. 80
CHAPTER 3 - GENERALIZED SOLUTIONS OF NONLINEAR PARTIAL DIFFERENTIAL
EQUATION~ ............................................ 83
§ 3.1 Explicit computations for shock wave solutions of systems
in nonconservative form ................................. 83
§ 3,2 Discontinuous solutions of the Cauchy problem for a system
in nonconservative form ................................. 89
§ 3.3 A new formulation of the equations of Hydrodynamics .... 106
§ 3.4 Jump formulas for shock waves in Elasticity and Elastoplasticity
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
§ 3.5 Existence-uniqueness for semilinear hyperbolic systems with
irregular Cauchy data .................................. 121
§ 3.6 Existence-uniqueness for a nonlinear parabolic equation with
irregular Cauchy data .................................. 133
APPENDIX I - Systems used by engineers for numerical simulations of
collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
APPENDIX 2 - Numerical tests in a system in nonconservative form. 148
APPENDIX 3 - Numerical tests in fluid dynamics ................... 157
APPENDIX 4 - Numerical tests in models of elastoplasticity ....... 175
APPENDIX 5 - Semilinear hyperbolic systems with irregular
coefficients and systems of equations in Acoustics..190
BIBLIOGRAPHIC NOTES .............................................. 198
REFERENCES ....................................................... 201
ALPHABETICAL INDEX ............................................... 212